Turbulent scattering for radars: A summary

Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

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Turbulent scattering for radars: A summary Franz-Josef Lübken n Leibniz-Institute of Atmospheric Physics at Rostock University, Schlossstraße 6, 18225 Kühlungsborn, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 25 June 2013 Received in revised form 23 October 2013 Accepted 24 October 2013 Available online 1 November 2013

In this paper some classical concepts regarding scattering of radio waves on turbulent structures in the ionosphere are summarized. Spectral representations according to Batchelor and Driscoll & Kennedy are compared and the role of various potential tracer gradients is elucidated. Systematic similarities and differences in the representation of the impact of these tracers on scatter intensity are investigated. The importance of turbulence and background parameters for radar volume reflectivities is discussed. This study highlights the importance of measuring these parameters as completely and reliably as possible when interpreting the strength of backscattered radar signal in terms of turbulent and atmospheric background parameters. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Radar scattering Turbulence Polar mesosphere summer echoes (PMSE) Polar mesosphere winter echoes (PMWE)

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variance dissipation rates and background gradients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Variance dissipation rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Background gradient for neutral density fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Background gradient for ion and electron density fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Background gradient for refractive index fluctuations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Spectral forms for Φϑ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Batchelor spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Driscoll & Kennedy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Comparison of spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Theories involving turbulence acting on electrons, ions, and ice particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Relationship among ΦnðkÞ, ΦNeðkÞ and ΦΔNe=NeðkÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Compilation of most important equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction Radar backscatter from electron density variations in the ionosphere caused by turbulence has been treated extensively in the past (see, for example, review by Hocking, 1985). Various representations

n

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1 2 2 2 2 3 3 3 3 4 4 5 5 6 6 6 6 7 7

of relevant turbulent and background parameters are available in the literature, depending on the context. For example, radar volume reflectivities (η) are compared with in situ measurements of relative electron density fluctuations. In this paper the main concepts of turbulent scattering for radars are summarized and some systematic similarities between various representations are highlighted. The background of this study is to identify sensitivities of η to relevant parameters such as Schmidt number (Sc), turbulent energy dissipation rates (ε), and mean electron number density (N e ). This summary

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F.-J. Lübken / Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

typically applies for VHF radars with a frequency of 50 MHz (λ/2¼3 m) but also at higher frequencies like the 224 MHz VHF EISCAT radar.

η ¼ 8π k4  C  X n  f ðkÞ ¼ 8π k4  C  X ϑ  α2ϑ  f ðkÞ

2. General We assume that turbulence in the atmosphere causes fluctuations of a passive tracer ϑ, the spectral distribution of which can be described by the three-dimensional power spectrum Φϑ ðkÞ relevant for radar scattering, or the one-dimensional power spectrum V ϑ relevant for, e.g., in situ measurements by rockets. Examples of such tracers are ΔN (neutral density), or ΔN e =Ne (relative electron density). Three-dimensional and one-dimensional power spectra are related through Z 1 V ϑ ðkÞ ¼ Φϑ ðkÞ  2π k  dk ð1Þ kz

Φϑ ðkÞ ¼

1 dVðkÞ  2π k dk

ð2Þ

where in the one-dimensional power spectrum kz is replaced by ω=vR if spectra are measured as a function of frequency (ω) when the rocket traverses the atmosphere with velocity vR. The functions ΦðkÞ are described by theoretical models which involve turbulence theory and background parameters. As has been shown by Tatarskii (1961) the cross section for radar scattering is given by ds 4 ¼ 2π k  V  Φn ðkÞ dΩ

ð3Þ

where Φn is the three-dimensional power spectral density of refractive index fluctuations. Integrating over a sphere and dividing by the measurement volume V gives the volume reflectivity:  Z  1 ds 4 ηðkÞ ¼ ð4Þ  dΩ ¼ 8π 2  k  Φn ðkÞ V dΩ Since backscattering mainly comes from scales at half the radar wavelength, i.e., at κ ¼ 2  2π =λ (Bragg condition) some authors present equations for κ ¼ 4π =λ ¼ 2  k instead of k. This leads to a modified version of Eq. (4) for κ :

ηðkÞ ¼

π2 2

κ 4 Φn ðkÞ

ð5Þ

This equation has been derived and applied by several authors, e.g., Ottersten (1969), Hocking (1985), and Rapp et al. (2008). In the ionosphere the main cause of refractive index variations is the fluctuation of electron number densities. In reality, relative electron density fluctuations are frequently measured. More generally, we consider a passive tracer where the models for Φn and Φϑ differ by a factor α2ϑ (see later):

Φn ðkÞ ¼ α2ϑ  Φϑ ðkÞ

ð6Þ

It turns out that turbulence models of ΦðkÞ are proportional to a quantity called ‘variance dissipation rate’. For the refractive index this quantity is labeled X n and therefore

Φn ðkÞ ¼ C  X n  f ðkÞ

ð7Þ

where C is a parameter which depends only on ε (and not on k). Combining with Eq. (6) introduces the variance dissipation rate for the passive tracer X ϑ :

Φϑ ðkÞ ¼ C 

depends on the wavenumber, for example X n (k), as will be shown below (Eq. (32)). For the radar volume reflectivity (Eq. (4)) this yields

Xn

αϑ 2

 f ðkÞ ¼ C  X ϑ  f ðkÞ

with X ϑ ¼

Xn

αϑ 2

ð8Þ

It should be noted that variance dissipation rates are normally considered to be constant, i.e., that they do not depend on the wavenumber. Examples of X e or X ΔN=N are introduced below. But there may also be cases where the variance dissipation rate

ð9Þ

For the concrete case of relative electron density fluctuations the theoretical models for ΦðkÞ differ by a factor of α2ω (see Appendix A):

Φn ðkÞ ¼ α2ω  ΦΔNe =Ne ðkÞ

ð10Þ

with 1  ωP  2 2 π r e ¼ 2  Ne 2 ω k

αω ¼ 

ðωP ¼ plasma frequencyÞ

ð11Þ

Applying Eq. (8) for this case gives

ΦΔNe =Ne ðkÞ ¼ C 

Xn

α2ω

 f ðkÞ ¼ C  X~n  f ðkÞ

ð12Þ

with X~n ¼ X n =α2ω . For the radar volume reflectivity (Eq. (4)) we get

η ¼ 8π k4  C  X n  f ðkÞ ¼ 8π k4  C  X~n  α2ω  f ðkÞ

ð13Þ

3. Variance dissipation rates and background gradients 3.1. Variance dissipation rates The variance dissipation rate X ϑ can be derived from the condition that the destruction of variance by molecular diffusion and the production by turbulence must be equal (Tatarskii, 1961): X ϑ ¼ f α Dϑ ðgrad ϑ′Þ2 ¼ f α K ϑ ðgrad ϑ Þ2

ð14Þ

where f α ¼ 2 is a constant (Lübken, 1992, and references therein). The quantity K ϑ can be expressed by the Prandtl number Pr¼K m =K ϑ and a classical expression for the eddy diffusion coefficient K m ¼ Ri  ε=ω2B : Kϑ ¼

Ri  ε Pr  ω2B

ð15Þ

where Ri is the Richardson number, ε is the turbulent energy dissipation rate, and ωB is the Brunt–Väisälä frequency (see Lübken, 1992, and references therein for a discussion of this equation). There are conceptual differences between K ϑ and ε since diffusion depends on the range of scales considered whereas energy dissipation does not. It should be noted that some background parameters are needed when applying this equation (e.g., Ri and ωB ) which are measured only with limited accuracy. Further uncertainties are introduced when specifying spatial and temporal scales being appropriate for determining the background relative to the turbulence field (see, for example, Gibson-Wilde et al., 2000, for a discussion on this topic). The background gradient of the passive tracer is sometimes also written as grad ϑ  M ϑ . Inserting into Eq. (14) gives Xϑ ¼ f α 

with

Ri  ε  M 2ϑ ¼ C ε  M 2ϑ Pr  ω2B

Cε ¼ f α 

Ri  ε Pr  ω2B

ð16Þ

ð17Þ

3.2. Background gradient for neutral density fluctuations For relative neutral density fluctuations the background gradient is given by (Lübken, 1992)

ω2 δN=N 1 1  M ΔN=N ¼  ¼ B HN γ Hp g δz

ð18Þ

F.-J. Lübken / Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

3

applying Eq. (18). Therefore

Inserting into Eq. (16) gives  2 2 ωB X ΔN=N ¼ C ε  g

ð19Þ

which depends on ε and on background parameters. Using f α ¼ 2, Ri¼ 0.81, ε ¼0.1 m2/s3, Pr¼1, ωB ¼ 0:02=s; g ¼ 9:81 m=s2 results in X ΔN=N ¼ 6:73  10  7 =s which is in the same order of magnitude as has been measured in rocket flights (X ΔN=N ¼ 10  8 …10  6 =s) (Lübken et al., 2006).

M~ e ¼ N e  f H

ð24Þ

and 2 X e ¼ C ε  M~ e ¼ C ε  ðN e  f H Þ2

ð25Þ

We note the following identity: F e  f H ¼ ω2B =g

ð26Þ

3.3. Background gradient for ion and electron density fluctuations

3.4. Background gradient for refractive index fluctuations

Thrane and Grandal (1981) have derived an expression for the variation of potential ion number densities, i.e., the difference between the ion number density of an air parcel which has been moved vertically, and the background ion number density. Since the spatial scales involved here are much larger than the Debye length we can assume charge neutrality and derive a similar relation for electron density variations:   δN e 1 1  δz ¼  ð20Þ He γHp Ne

The following part goes back to Hocking (1985). To be consistent with our nomenclature of background variations of an arbitrary tracer M ϑ and the nomenclature introduced by Rapp et al. (2008) we use M n for the background gradient of potential refractive index variations (labeled M e by Hocking) and M~ e for the background gradient of potential electron density variations. Since the refractive index depends only on electron density Ne (see Appendix A) the variation of refractive index with altitude is given by

The main idea of potential electron number density variations is sketched in Fig. 1. Note that the electron density scale height 1=H e ¼  ð1=Ne Þ  ðdN e =dzÞ may vary substantially in the D- and Eregion. By combining Eqs. (18) and (20) the altitude displacement δz can be eliminated to obtain the following relation between neutral density (N) and electron density (Ne) variations:

Mn 

δN N

¼ Fe 

δN e Ne

γ Hp ;

Fe ¼

HN

γ Hp He

1

ðH p ¼ pressure scale heightÞ

ð21Þ

1

Combining Eqs. (18) and (20) gives the expression derived by Hocking (1985) for the potential electron density gradient:  2  ωB 1 1 δN e ¼ Ne  þ  ð22Þ δz g He HN This implies that the derivation of the potential refractive index variation by Hocking (1985) is consistent with the derivation of potential ion density fluctuations proposed by Thrane and Grandal (1981). To be consistent with other publications and for simplification we write in Eq. (22) δNe =δz  M~ e and fH for the term in brackets:  2  ωB 1 1 1 1 þ  ð23Þ fH ¼ ¼  He γ  Hp g He HN altitude

electrons

neutrals

∂n ∂n ∂Ne ∂n ¼ ¼   M~ e ∂z ∂N e ∂z ∂N e

ð27Þ

Note that ∂N e is the difference between the electron density in the air parcel (moved by turbulence) and the mean background electron density. This is the reason for calling it ‘potential refractive index gradient’ (see Fig. 1). From Eq. (60) (Appendix A) we see that ∂n 1 2π r e 8π r e 2  re  λ ¼ 2 ¼ 2 ∂N e 2π κ k

ð28Þ

using κ ¼ 4π =λ. Inserting Eqs. (28) and (24) into Eq. (27) gives the required result for the background gradient of potential refractive index variation: 2π r e  M~ e ¼ 2  M~ e k  2  ωB 1 1 2π r e þ  ¼ 2  Ne  g He Hn k

Mn ¼

8π r e

ð29Þ

κ2

ð30Þ

¼ αω  f H

ð31Þ

and X n ¼ C ε  M 2n ¼ C ε ðαω  f H Þ2 ¼ C ε 



2π  r e k

2

2  Ne

2

 fH

ð32Þ

4. Spectral forms for Φϑ 4.1. Batchelor spectrum

background

~ 0.5%

~ 5%

number density

Fig. 1. Sketch of variations of neutral (green) and electron number densities (red) when an air parcel is moved vertically in a background atmosphere (blue). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Batchelor (1959) has derived a spectrum for a passive tracer ϑ in the viscous-convective subrange, i.e., for scales smaller than the 1=4 Kolmogorov microscale ηKol ¼ ðν3 =εÞp ffiffiffiffiffiand larger than the Batchelor scale ηBa ¼ ðν  D2 =εÞ1=4 ¼ ηKol = Sc (Sc ¼ ν/D ¼Schmidt number; D ¼molecular diffusion coefficient; ν ¼kinematic viscosity). For this range the Batchelor spectrum for a passive tracer ϑ is given by (see Rapp et al., 2008) ! 2 1 Xϑ Dϑ k Ba  exp Φϑ ¼  ð33Þ 2 γε 4π k γ ε  k which is normalized to yield Z 1 Z 1 Z 1 ′2 ΦBa ϑ ðkÞ dkx dky dkz ¼ ϑ ; 1

1

1

! k¼jk j

ð34Þ

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F.-J. Lübken / Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

Insertingpthe ffiffiffiffiffiffiffiffi rate of strain of Kolmogorov eddies given by

γ ε ¼  ð1=qÞ ε=ν (with q ¼2) gives ΦBa ϑ ¼

1 4π k

 2

2η Xϑ  exp  k γε  k Sc

2 Kol 2

Xϑ

 exp 

γε

ð35Þ

2η k Sc

2 Kol 2

ð36Þ

Ba ΦBa n ðkÞ ¼ Φϑ ðkÞ and X n ¼ X ϑ

ð37Þ

γε

 exp 

2η2Kol 2 k Sc

! ð38Þ

The variance dissipation rate from Eq. (16) is



2π r e 2

k

ð39Þ

 M~ e

2

which yields the following expression: 2η2 η ðkÞ ¼ 2π k  C ε  ðαω ðkÞ  f H Þ   exp  Kol k2 γε Sc 2

Ba

1

ð40Þ

Again, we make the important assumption that refractive index variations are passive tracers, e.g., D&K ΦD&K ðkÞ ¼ Φϑ ðkÞ and X n ¼ X ϑ n

ð41Þ

This is identical to Eq. (14) in Rapp et al. (2008). 4.2. Driscoll & Kennedy spectrum A general expression for the 3-dimensional power spectrum for a passive tracer at scales within the inertial subrange (and smaller) was given by Driscoll and Kennedy (1985) (D&K) and later applied by Giebeler (1995): Q2 A1  β  DðyÞ 4π

ð50Þ

which (in combination with Eq. (4)) gives the equation for the volume reflectivity for a D&K spectrum: ð51Þ

or, with k ¼ κ =2:

ηD&K ðkÞ ¼

π2 2

11=3 κ 4  Q 9=2 AX n ε  1=3 ηKol  DðyÞ

ð52Þ

X n ¼ C ε  ðαω  f H Þ2 ¼ f α 

Ri  ε  M 2n Pr  ω2B

ð53Þ

with the expression for αω and fH from Eqs. (63) and (23), respectively. 4.3. Comparison of spectra

!

Inserting Eqs. (39) and (29) for M n into Eq. (35) and then into Eq. (5) yields the desired equation for the radar volume reflectivity: ! pffiffiffiffiffiffi 2η2Kol 2 εν ~ 2 2 1 Ba 3 f α  q  Ri M  r e  3  exp  k η ðkÞ ¼ 8π ð42Þ Pr Sc ω2B e k

11=3  1=3 ΦD&K  ηKol  ϑ ðkÞ ¼ εϑ  ε

ð49Þ

The variance dissipation rate can be expressed as (see Eq. (16))

Ri  ε  M 2n ¼ C ε  M 2n Xn ¼ f α  Pr  ω2B ¼ C ε  ðα ω  f H Þ2 ¼ C ε 

ð48Þ

This gives the following expression for the D&K spectrum:

11=3 ηD&K ðkÞ ¼ 8π 2  k4  Q 9=2 AX n ε  1=3 ηKol  DðyÞ

and thereby Xn

Q2  A1  β ¼ A  Q 9=2  X ϑ 4π

11=3 Φϑ ðkÞ ¼ Q 9=2 AX ϑ ε  1=3 ηKol  DðyÞ

!

We now make the important assumption that refractive index variations are passive tracers, e.g.,

ηBa ðkÞ ¼ 2π k 

εϑ 

D&K

and for volume reflectivity:

ηBa ¼ 2π k 

!

(43) can be written as

Within spatial scales given by the Kolmogorov and Batchelor scales the Batchelor and D&K spectra are rather similar (see Fig. 2). The actual difference within these scales depends on the wavenumber, on ε, and on the Schmidt number Sc. The difference is typically less than 30–50%. Outside the exponential decay, i.e., for Sc ¼1, the ratio of volume reflectivities is given by   ηD&K ΦD&K 1 ¼ ¼ 2π  A  1 þ 2=3 ð54Þ Ba Ba η y Φ which equals 0.5411 for k ¼ 1=ηKol . Ratios as a function of wavenumber are shown in Fig. 3 for several Schmidt numbers and other parameters as in Fig. 2. As can be seen from this figure, spectra differ by typically a factor of two at scales between the Kolmogorov and Batchelor microscales.

ð43Þ

with

n o DðyÞ ¼ ðy  11=3 þ y  3 Þ  exp  A3ϑ ð32 y4=3 þ y2 Þ where y ¼ Q Q ¼ 2;

3=2

A3ϑ ¼

A1 ¼ α  Q 5=2 ;

ð44Þ

 ηKol  k and

α Q 2  Sc

α ¼ 0:83

ð45Þ ð46Þ

Comparing the one-dimensional spectrum from Driscoll & Kennedy with the one-dimensional power spectrum from Heisenberg in the inertial subrange results in the following expression for εϑ (Giebeler, 1995): 4π A εϑ ¼ X αβ ϑ

ð47Þ

(A ¼ 0:033  a2 , a2 ¼1.74) which shows that εϑ is proportional to X ϑ . Together with Eq. (46) a combination of constants appearing in Eq.

Fig. 2. Spectra of volume reflectivities for Batchelor (blue) and D&K (red) spectra for Ne ¼ 1  10  9/m3, dNe/dz¼ 1  105/m4, ε ¼0.1 m2/s3, and Sc ¼10, 100, 1000 for the solid, dashed, and dashed-dotted lines. Other ‘nominal’ parameters are as explained in the text. The vertical lines indicate the radar Bragg scale (black), the Kolmogorov microscale (red) and the Batchelor scales (blue). The green line shows  11=3 spectral slopes of k and k  3. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

F.-J. Lübken / Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

5

Fig. 4. Variance dissipation rates X n in [1/s] as a function of ε and Ne for a fixed value of dNe/dz¼1  105/m4 and ν ¼ 1 m2 =s. Fig. 3. Ratio of Batchelor to Driscoll & Kennedy power spectral densities as a function of wavenumber for Schmidt numbers of 10 (blue), 100 (green), and 1000 (red), respectively. Other parameters as in Fig. 2. The vertical lines indicate the Kolmogorov microscale (red) and the Batchelor scales (blue, green red) according to the Schmidt numbers given above. Note that the ratio is independent of Schmidt number if Sc b 1 and k o 1=ηBa . (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

5. Discussion The most important equations are compiled in Appendix B for easy application. 5.1. Sensitivities Volume reflectivity spectra depend on turbulence strength (ε) and variance dissipation rate (X n ) which is sometimes assumed to be constant (see Appendix B) or is derived from background parameters such as N e and dN e =dz. For a fixed wavenumber, for example k ¼ 2  2π =λ (Bragg condition) the expression for η depends on ε, Sc(ν), and on X n which in turn depends on various background parameters (if not assumed to be constant):

ηD&K ¼ f ðε; ηKol ; Sc; X n Þ

ð55Þ

X n ¼ f ðε; N e ; dN e =dz; Ri; Pr; ωB ; H N Þ

ð56Þ

Note that the kinematic viscosity ν enters through ηKol and Sc. Kinematic viscosity increases exponentially with height but is taken as ν ¼ 1 m2 =s in the major part of this paper, representative for an altitude of approx. 85 km. In Fig. 4 the variation of X n as a function of ε and Ne is shown for a fixed value dNe =dz ¼ 1  105 =m4 and other standard values such as Ri¼0.81, Pr¼1, ωB ¼ 0:02=s, and HN ¼ 7 km. Obviously, X n (and therefore ηD&K ) can easily vary by several orders of magnitude even for a fixed electron density gradient and other background parameters. In order to compare radar measurements of η with theory it is therefore necessary to measure the neutral and plasma structure of the atmosphere as well as turbulent parameters as accurately as possible. Further parameters get involved when calculating volume reflectivities (see above). In Fig. 5 we show ηD&K at the Bragg wavenumber as a function of Schmidt number and electron number densities for fixed values of dNe/dz¼1  105/m4 and ε ¼0.1 m2/s3 (Ri, Pr, ωB , and HN as above). Part of the motivation for this study came from the question if radar echoes observed in the lower mesosphere (known as polar mesosphere winter echoes, PMWE) require the existence of charged aerosols, or not (e.g., Lübken et al., 2006; Zeller et al., 2006; Morris et al., 2011). In Fig. 6 we show turbulence energy dissipation rates ε(z) needed to achieve a fixed volume reflectivity of ηD&K ¼ 1  10  15 =m (typical for PMWE) and various Schmidt numbers

Fig. 5. Volume reflectivities in 1/m as a function of Sc and Ne for fixed values of dNe/ dz¼ 1  105/m4, ε¼ 0.1 m2/s3, and ν ¼ 1 m2 =s.

Fig. 6. Energy dissipation rates required to achieve a fixed volume reflectivity of η ¼1  10  15/m for various Schmidt numbers (see inlet). Equivalent particle radii rd (in nm) are also given using Sc ¼ 6:5  r 2d from Lübken et al. (1998). The electron density profile was taken according to IRI but multiplied by a factor of 10. Electron density gradient profile was also taken according to IRI. Other constants like in previous plots.

(see inlet). Equivalent particle radii are also given using Sc ¼ 6:5  r 2d from Lübken et al. (1998) (rd ¼particle radius in nm). The background electron profile was taken according to the international reference ionosphere (IRI) but multiplied by a factor of 10 to account for enhanced ionization typically present during PMWE. As can be seen from Fig. 6 huge energy dissipation rates are needed above 80–85 km if there are no charged ice or dust particles present in the atmosphere, i.e., if Sc¼1. In the lower and middle mesosphere, however, energy dissipation rates are moderate and nearly independent of Sc. This highlights the result from Lübken et al. (2006) that charged

6

F.-J. Lübken / Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

particles are not imperatively needed to explain PMWE in the middle and lower mesosphere. Of course, the existence of charged dust particles in the mesosphere cannot be excluded from these considerations. Indeed, there are experimental indications that charged particles are present in PMWE (Havnes et al., 2010). As has been mentioned above, the results shown in Fig. 6 strongly depend on the background electron density profile.

5.2. Theories involving turbulence acting on electrons, ions, and ice particles In this paper electrons are treated as passive tracers where the effect of charged ice or dust particles is considered through an enhanced Schmidt number because the diffusivity of free electrons is hampered by ‘heavy’ particles through an ambipolar electric field which finally leads to polar mesosphere summer echoes (PMSE) (and references therein Kelley et al., 1987; Cho and Kelley, 1993; Lübken et al., 1998; Rapp and Lübken, 2003). A more rigorous treatment considers turbulence acting on all three constituents (electrons, ions, and charged aerosols) and takes into account various diffusion modes in the plasma (Hill, 1978; Hill et al., 1999; Rapp and Lübken, 2003). In a recent paper Varney et al. (2011) have revisited these theories and studied in particular plasma diffusion modes at small scales as well as the role of electron density in determining radar reflection. They arrive at an equation for ηBa (k) similar to Eq. (42) but basically replacing M~ e by Mdust. It is somewhat questionable if the idea of turbulent mixing on a smooth background gradient can be applied to ice particles since they are subject to microphysical processes which may cause small scale layering (see, for example, NLC layers shown in Kaifler et al., 2013). Furthermore, Mdust is difficult to measure which introduces uncertainties. We note that the difference between the classical representation presented in this paper and the full diffusion model discussed by Varney et al. (2011) is relevant only for high electron densities, more precisely for small Λ values (Λ ¼ N d =N e Þ (Nd ¼number density of charged dust particles). We will discuss the implications of the results of Varney et al. in a future paper. Another important tool to study radar scattering from turbulence comes from direct numerical simulations (DNS). For example, Fritts et al. (2012) (and references therein) present a detailed discussion of radar backscatter from turbulence created by Kelvin–Helmholtz instabilities associated with gravity wave breaking. Such studies allow to compare theoretical models (as presented in this paper) with numerical simulations and to investigate, for example, the importance of background conditions, although only for limited scenarios given by the boundary conditions of the DNS.

5.3. Summary and conclusion It has been shown that the spectral forms of volume reflectivities according to Batchelor and Driscoll & Kennedy, respectively, are similar for scales between the Batchelor and Kolmogorov scales. Apart from turbulence intensity, the amount of backscattering is critically determined by the background gradient of refractive index variations, which in turn depends on various background parameters, most importantly on electron number density. Obviously, it is very important to measure background parameters as completely and reliably as possible when interpreting radar volume reflectivities in terms of turbulent parameters. In particular, the electron density profile should be measured with great care. We plan to perform simultaneous and colocated radar, lidar, and in situ measurements of all relevant parameters during a PMWE event at ALOMAR and the Andøya Rocket Range in the near future.

Acknowledgments I gratefully acknowledge discussions with Irina Strelnikova on the Varney et al. paper. Appendix A. Relationship among Φn ðkÞ, ΦNe ðkÞ and ΦΔNe =Ne ðkÞ As can be shown from Maxwell equations the refractive index for wave propagation is given by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω 2 1 ωP 2 P n ¼ 1  1  ð57Þ 2 ω ω where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2  N e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ωP ¼ ¼ r e  4π c 2  N e ε ○  me

ð58Þ

is the plasma frequency which has been rewritten applying the definition of the classical electron radius re ¼

1 e2 ¼ 2:8179  10  15 m 4πε○ me c2

ð59Þ

Inserting this relation and ω2 ¼ ð2π Þ2  c2 =λ yields: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 re  Ne  λ 1 re  Ne  λ n ¼ 1  1 2 π π

2

into Eq. (57)

ð60Þ

as quoted, for example, by Hocking (1985). The variation of the refractive index is then given by (see Eq. (57)) ∂n 1 1 ω 2 Δn ¼ ΔN e ¼   P  ΔN e ¼ α~ ω  ΔNe ð61Þ ∂N e Ne 2 ω where we have introduced the constant 1 1 ω 2 α~ ω ¼   P af ðNe Þ Ne 2 ω

ð62Þ

from Giebeler (1995) which is independent of electron density Ne. For relative electron density fluctuations we get

Δn ¼ α ω 

ΔN e

ð63Þ

Ne

αω ¼ α~ ω  N e ¼

8π  r e

κ2

 Ne ¼

2π  r e 2

k

 Ne

ð64Þ

Power spectral densities of electron density fluctuations are normalized such that Z 1 Z 1 Z 1 ΦNe ðN e Þd3 k ¼ ðΔNe Þ2 ð65Þ 1

1

1

Similar equations hold for relative electron density fluctuations and refractive index fluctuations. From the equivalent equations we see, for example, that ðΔnÞ2 ¼ α~ 2ω  ðΔNe Þ2 

ΔN N

2

 ¼ F 2e 

ΔN e

2

Ne 

¼ F 2i 

ð66Þ

ΔN i

ð67Þ

2

Ni

ð68Þ

which leads to the following relations for power spectral densities:

Φn ðkÞ ¼ α~ 2ω  ΦNe ðkÞ

ð69Þ

Φn ðkÞ ¼ α2ω  ΦΔNe =Ne ðkÞ

ð70Þ

ΦΔN=N ¼ F 2e  ΦΔNe =Ne

ð71Þ

F.-J. Lübken / Journal of Atmospheric and Solar-Terrestrial Physics 107 (2014) 1–7

ΦNe ¼ N 2e  ΦΔNe =Ne

ð72Þ

X ΔNe =Ne ¼

and corresponding relations between the variance dissipation rates (see Appendix B). with Appendix B. Compilation of most important equations

ð73Þ

ηD&K ðkÞ ¼ 8π 2  k  Q

AX n ε

η

 1=3 11=3 Kol

 DðyÞ

ð74Þ 2

with D(y) from Eq. (44) and Q¼ 2, A ¼ 0:033  a2 , a ¼1.74, A3ϑ ¼ α=Q 2  Sc, α ¼0.83. Note that in Lübken et al. (2006) a (wrong) factor of 16π2 instead of 8π2 was given in Eq. (74) which stems from Giebeler (1995, p. 63), where he states a factor of 4π which should be 2π. Three-dimensional power spectra:

ΦðkÞ ¼

ηðkÞ 4 8π 2  k

ð75Þ

Variance dissipation rates from background conditions: X n ¼ C ε  ðα ω  f H Þ2  2π r e

¼ Cε 

k

2

ð76Þ 2

 Ne  f H 

X e ¼ C ε  ðN e  f H Þ2 ¼

2  Xn

αω

 Xn

2π  r e 

X ΔN=N ¼ C ε 

ω2B

ð79Þ

2 ð80Þ

g

X e ¼ N 2e  X ΔNe =Ne

ð81Þ

X ΔN=N ¼ F 2e  X ΔNe =Ne ¼

¼

ð78Þ

!2

2

k

¼

ð77Þ

Ne

F 2e

 2

Ne

k

F 2e N2e

 Xe

ð82Þ

!2

2

 Xn

2π  r e

ð83Þ

with Cε ¼ f α 

αω ¼

Ri  ε Pr  ω2B

2π r e k

2

 fH ¼

Fe ¼

ω2B g

ð84Þ

 Ne

þ

1 1  He HN

ð85Þ  ¼

1 1  He γ  Hp

ω2B 1 g



fH

ð86Þ

ð87Þ

and f α ¼ 2, re ¼2.8179  10  15 m, and Pr ¼1, Ri¼ 0.81 as nominal values. Alternatively, X ΔN=N from rocket measurements X ΔN=N  10  6 …10  8 =s

 25…100

ð89Þ

ð90Þ

 X ϑ ¼ Cε 

αω αϑ

2

2

 fH

ð91Þ

References

Volume reflectivities for a Driscoll & Kennedy spectrum: 9=2

 X ΔN=N  1  10  5 =s

Comparing the two equations for X n (Eqs. (76) and (8)) gives the following expression for X ϑ :

Volume reflectivities for a Batchelor spectrum: ! 2η2 X ηBa ðkÞ ¼  2π k  n  exp  Kol k2 γε Sc pffiffiffiffiffiffiffiffi with γ ε ¼  ð1=qÞ ε=ν, q  2, ηKol ¼ ðν3 =εÞ1=4 .

4

1 F 2e

1 F 2e

7

ð88Þ

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